In the next section, we construct an ensemble of \(N=4M-4\)  measurement vectors in \(\mathbb {C}^M\)  which yield injective intensity measurements. This is the second known injective ensemble of this size (the first is due to Bodmann and Hammen [1]}), and it is conjectured to be the smallest-possible injective ensemble [2]}.
The same conjecture suggests that \(4M-4\)  generic measurement vectors yield injectivity (that is, there exists a measure-zero set of ensembles of \(4M-4\)  vectors such that every ensemble of \(4M-4\)  vectors outside of this set yields injectivity).
The following summarizes what is currently known about the so-called “\(4M-4\)  conjecture”:
